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arXiv:2406.17441

A Matrix Product State Model for Simultaneous Classification and Generation

Alex Mossi, Bojan Žunkovič, Kyriakos Flouris

Published Jun 25, 2024• 1 citation
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Abstract

Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Initially developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Drawing inspiration from these quantum methods, specifically the Matrix Product States (MPS), we apply them in a classical machine learning setting. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them effective in a supervised learning framework. Here, we present an MPS model, in which the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. In addition, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.

DOI
10.48550/arxiv.2406.17441

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Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Initially developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Drawing inspiration from these quantum methods, specifically the Matrix Product States (MPS), we apply them in a classical machine learning setting. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them effective in a supervised learning framework. Here, we present an MPS model, in which the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. In addition, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
Abstract Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Originally developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them suitable for various machine learning tasks within the quantum realm. Here, we present a matrix product state (MPS) model, where the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. Additionally, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.
Optimizing hyperparameters in deep learning models remains a significant bottleneck in the development and deployment of effective neural networks. Traditional methods, such as grid search and random search, often struggle with the exponentially growing search space, leading to substantial computational costs and limited exploration of potentially optimal configurations. This paper proposes a novel approach to deep learning hyperparameter optimization that leverages the principles of quantum-inspired optimization, specifically employing tensor network methods, notably Matrix Product States (MPS), to efficiently approximate the complex optimization landscape. Our method utilizes quantum-inspired sampling techniques to guide the exploration, effectively mimicking the probabilistic nature of quantum measurements. We demonstrate the feasibility and potential advantages of this approach through theoretical analysis and a conceptual framework. The core of our method lies in representing the hyperparameter space as a tensor network, allowing for efficient calculations of the objective function (e.g., validation loss) without exhaustively evaluating every possible combination. This significantly reduces the computational burden compared to conventional methods. The paper outlines a detailed algorithm and discusses key considerations for implementation, including tensor network construction, sampling strategies, and convergence criteria. We propose a framework for iteratively refining the tensor network representation based on observed validation performance, ultimately converging towards a high-performing hyperparameter configuration. The theoretical underpinning draws upon concepts from quantum mechanics, particularly the idea of measurement and probability distribution, to drive the optimization process. We believe this quantum-inspired approach offers a promising avenue for accelerating deep learning hyperparameter tuning and achieving state-of-the-art performance.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
The tensor network (TN) has recently triggered extensive interests in developing machine learning models defined in quantum Hilbert space. Here, we propose a generative TN classification (GTNC) model for supervised machine learning. The strategy is first to map the classical data onto the states in a many-body Hilbert space and then to capture these states with tensor network schemes. We adopt the TN in the form of matrix product states as an example to implement GTNC where the testing images are classified by comparing the fidelities between different states. Our results show that GTNC has a very impressive performance on benchmark datasets in comparison to several well-known machine learning models. The advantage of GTNC is reflected from the facts that the samples are naturally clustering in the many-body Hilbert space, and it relies much less on hyperparameters. These characters make GTNC an adaptive and universal quantum-inspired method, which would have important applications in quantum computation and quantum information.
Matrix product states (MPSs), a tensor network designed for one-dimensional quantum systems, were recently proposed for generative modeling of natural data (such as images) in terms of the ``Born machine.'' However, the exponential decay of correlation in MPSs restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the tree tensor network (TTN), which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the two-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing graphical processing units. We apply our model to random binary patterns and the binary MNIST data sets of handwritten digits. We show that the TTN is superior to MPSs for generative modeling in keeping the correlation of pixels in natural images, as well as giving better log-likelihood scores in standard data sets of handwritten digits. We also compare its performance with state-of-the-art generative models such as variational autoencoders, restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of tensor network states in machine learning problems.